P3: DifferentiationEdexcel International A Level Maths: Topic test

1. 1
   The function $f$ is defined by $f(x)=3e^{2x}+4\sin\frac{x}{2}$, where $x$ is in radians.

   (a)

   Which of the following is $f'(x)$?

   [1 mark]

   • A$6e^{2x}+8\cos\frac{x}{2}$
   • B$6e^{2x}+2\cos\frac{x}{2}$
   • C$3e^{2x}+2\cos\frac{x}{2}$
   • D$6e^{2x}-2\cos\frac{x}{2}$

   (b)

   Find the exact value of $f'(\pi)$.

   [1 mark]

   • A$6e^{2\pi}+2$
   • B$6e^{2\pi}-2$
   • C$6e^{\pi}$
   • D$6e^{2\pi}$

   (c)

   Find an equation of the tangent to the curve $y=f(x)$ at the point where $x=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The value, in pounds, of a machine $t$ years after it was bought is modelled by $V=18000e^{-0.2t}$ for $t\ge0$.

   (a)

   Find the rate of change of $V$, in pounds per year, when $t=0$.

   [1 mark]

   • A$-3600$
   • B$3600$
   • C$-0.2$
   • D$-18000$

   (b)

   After how many years does the model predict that the value has halved?

   [1 mark]

   • A$-3.47$
   • B$0.139$
   • C$3.47$
   • D$2.5$

   (c)

   Describe the long-term behaviour of $V$ predicted by the model, and give one reason why this may be unrealistic.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\dfrac{4x}{\sqrt{3x+1}}$ for $x>-\frac13$.

   (a)

   Show that $\dfrac{dy}{dx}=\dfrac{6x+4}{(3x+1)^{\frac32}}$.

   [3 marks]

   (b)

   Find an equation of the normal to $C$ at the point where $x=1$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=e^{2x}\cos x$ for $0\le x\le\pi$.

   (a)

   (i) Find $\dfrac{dy}{dx}$.
   (ii) Show that the $x$-coordinate of the stationary point of $C$ satisfies $\tan x=2$.
   (iii) Find the coordinates of the stationary point, giving each coordinate to 3 significant figures.

   [6 marks]

   (b)

   (i) Show that $\dfrac{d^2y}{dx^2}=e^{2x}(3\cos x-4\sin x)$.
   (ii) Hence show that $\dfrac{d^2y}{dx^2}-4\dfrac{dy}{dx}+5y=0$.
   (iii) Use the result in (ii) to show that the stationary point of $C$ is a maximum.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $C$ has equation $x=\cos2y$ for $0<y<\frac{\pi}{2}$.

   (a)

   Which of the following is $\dfrac{dy}{dx}$?

   [1 mark]

   • A$\dfrac{1}{2\sin2y}$
   • B$-2\sin2y$
   • C$-\dfrac{1}{2\sin2y}$
   • D$-\dfrac{1}{\sin2y}$

   (b)

   Find the gradient of $C$ at the point where $y=\frac{\pi}{12}$.

   [1 mark]

   • A$-1$
   • B$1$
   • C$-1.93$
   • D$-2$

   (c)

   Find the coordinates of the point on $C$ at which the gradient is $-\frac12$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $g$ is defined by $g(x)=\dfrac{\cos x}{1+\sin x}$ for $0\le x\le\pi$.

   (a)

   Which of the following is $g'(x)$?

   [1 mark]

   • A$\dfrac{1}{1+\sin x}$
   • B$-\dfrac{\sin x}{1+\sin x}$
   • C$-\dfrac{1}{(1+\sin x)^2}$
   • D$-\dfrac{1}{1+\sin x}$

   (b)

   Find the value of $g'\left(\frac{\pi}{6}\right)$.

   [1 mark]

   • A$\dfrac23$
   • B$-\dfrac23$
   • C$-2$
   • D$-\dfrac{2}{2+\sqrt3}$

   (c)

   Find an equation of the normal to the curve $y=g(x)$ at the point where $x=0$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has equation $y=x\,5^{x}$ for $x\ge0$.

   (a)

   Find $\dfrac{dy}{dx}$.

   [3 marks]

   (b)

   Show that the tangent to $C$ at the point where $x=1$ passes through the point $(0,\,-5\ln5)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A drug is taken orally at time $t=0$. The concentration $C$ micrograms per millilitre of the drug in the blood, $t$ hours after it is taken, is first modelled by $C=16e^{-0.5t}$. To improve the model, it is then modelled by $C=20te^{-0.5t}$. Both models are for $t\ge0$.

   (a)

   (i) Explain why the first model is unsuitable at $t=0$.
   (ii) For the improved model, find $\dfrac{dC}{dt}$.
   (iii) Hence find the time at which the improved model predicts the greatest concentration, and the value of this greatest concentration to 3 significant figures.

   [6 marks]

   (b)

   (i) For $t>0$, find the time at which the two models predict the same concentration, and give this concentration to 3 significant figures.
   (ii) For the improved model, find the rate of change of the concentration when $t=4$, and state what this shows.
   (iii) Using the first model, find the time at which the concentration falls to 2 micrograms per millilitre, giving your answer to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions